The existence of perfect Mendelsohn designs with block size 7 Original Research Article

Necessary conditions for existence of a (v,k,λ) perfect Mendelsohn design (or PMD) are v ⩾ k and λv(v − 1) ≡ 0 mod k. When k = 7, this condition is satisfied if v ≡ 0 or 1 mod 7 and v ⩾ 7 when λ ≢ 0 mod 7 and for all v ⩾ 7 when λ ≡ 0 mod 7. Bennett, Yin and Zhu have investigated the existence problem for k = 7, λ = 1 and λ even; here we provide several improvements on their results and also investigate the situation for λ odd. We reduce the total number of unknown (v,7,λ)-PMDs to 36,31 for λ = 1 and 5 for λ > 1. In particular, v = 294 is the largest unknown case for λ = 1, and the only unknown cases for λ > 1 are for v = 42, λ ∈ [2,3,5,9] and v = 18, λ = 7.